Page:Calculus Made Easy.pdf/130

(3) A line of length $p$ is to be cut up into $4$ parts and put together as a rectangle. Show that the area of the rectangle will be a maximum if each of its sides is equal to ${\tfrac {1}{4}}p$.

(4) A piece of string $30$ inches long has its two ends joined together and is stretched by $3$ pegs so as to form a triangle. What is the largest triangular area that can be enclosed by the string?

(5) Plot the curve corresponding to the equation

$y={\frac {10}{x}}+{\frac {10}{8-x}}$;

also find ${\dfrac {dy}{dx}}$, and deduce the value of $x$ that will make $y$ a minimum; and find that minimum value of $y$.

(6) If $y=x^{5}-5x$, find what values of $x$ will make $y$ a maximum or a minimum.

(7) What is the smallest square that can be inscribed in a given square?

(8) Inscribe in a given cone, the height of which is equal to the radius of the base, a cylinder (a) whose volume is a maximum; (b) whose lateral area is a maximum; (c) whose total area is a maximum.

(9) Inscribe in a sphere, a cylinder (a) whose volume is a maximum; (b) whose lateral area is a maximum; (c) whose total area is a maximum.